Q.Find the distance between the parallel lines $3x + 4y = 9$ and $6x + 8y = 15$. (First check that the two equations really do represent parallel lines with a common $A:B$ ratio.)

[!TLDR] Rescale the second equation to share the same $A,B$ as the first, then apply the parallel-line distan …

$3x+4y-9=0$ has $A=3,B=4,C_1=-9$. The second equation $6x+8y-15=0$ has coefficients exactly double those of the first except for the constant; dividing throughout by $2$ rescales it to the SAME $A,B$: $3x+4y-7.5=0$, i.e. $C_2=-7.5$. (This rescaling step is essential — the distance formula requires the two equations to share identical $A,B$, not merely proportion …

Divide the second equation through by the common scale factor (here $2$) so both lines share identical $A,B$, t …

Applying the formula directly to $3x+4y-9=0$ and $6x+8y-15=0$ without first rescaling — i.e. using $A=3,B=4$ from one and $C_2=-15$ straight from the other …